of heat energy due to mechanisms such as north-south boundary currents and mixing due to eddies that are shed
The ocean plays a crucial role in the earth's climate system, and an improved understanding of that role will be aided greatly by from ocean currents (Robinson [18]). In middle latitudes, high-resolution simulations of global ocean circulation over periods oceanic eddies typically have widths on the order of 200 of many years. For such simulations the computational requireto 300 km. However, eddies in the higher latitudes typically ments are extremely demanding and maximum efficiency is essenhave widths of about 50 km or even less, and boundary tial. However, the governing equations typically used for ocean currents can have widths of roughly this magnitude. A modeling admit wave velocities having widely varying magnitudes, and this situation can create serious problems with the efficiency numerical model that resolves such features over long of numerical algorithms. One common approach to resolving these times will thus pose some very large computational deproblems is to split the fast and slow dynamics into separate submands.
problems. The fast motions are nearly independent of depth, and
In such a model, the most straightforward time discretiit is natural to try to model these motions with a two-dimensional zation would be an explicit method. However, such a system of equations. These fast equations could be solved with an implicit time discretization or with an explicit method with short method would suffer from a severe restriction on the allowtime steps. The slow motions would then be modeled with a threeable time step, due to the nature of the wave motions dimensional system that is solved explicitly with long time steps that that are typically found in large-scale models of ocean are determined by the slow wave speeds. However, if the splitting is circulation. In existing models it is commonplace to elimiinexact, then the equations that model the slow motions might nate sound waves by various means. The remaining dynamactually contain some fast components, so the stability of explicit algorithms for the slow equations could come into doubt. In this ics then include several classes of waves; prominent among paper we discuss some general features of the operator splitting these are external gravity waves (i.e., surface gravity problem, and we then describe an example of such a splitting and waves) and internal gravity waves. External motions are show that instability can arise in that case.